Delayed dynamic-feedback controller design for multi-frequency vibration suppression

Adam Peichl; Adrian Saldanha; Tom\'a\v{s} Vyhl\'idal; Wim Michiels

REVIEW 2 major objections 5 minor 4 references

Delayed dynamic-feedback controller design for multi-frequency vibration suppression

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read By adding fixed output delays and a dynamic feedback block, a single actuator can place transmission zeros at four prescribed frequencies while simultaneously minimizing the closed-loop spectral abscissa.

desk verdict A sound but incremental extension of the authors' delayed-feedback framework to dynamic controllers, with the main soft spot being the unexamined invertibility assumptions in the constraint elimination. read the letter →

arxiv 2505.16939 v1 pith:WOMMB3SB submitted 2025-05-22 eess.SY cs.SYmath.OC

Adrian Saldanha , Adam Peichl , Wim Michiels , Tomáš Vyhlídal This is my paper · ORCID

classification eess.SYcs.SYmath.OC MSC 93C2393B55

keywords vibrationsuppressiontime-delaysystemsdelay-differentialalgebraicequationstransmission-zeroplacementspectralabscissaoptimizationnon-collocatedcontroldynamicoutputfeedbackmulti-frequencyharmonicdisturbance

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper presents a controller-design method for canceling harmonic vibrations at a target point when the actuator cannot be placed there. In this non-collocated setting, each targeted frequency requires two free controller parameters, and a static feedback loop rarely has enough. The authors show that adding several fixed output delays and a low-order dynamic controller creates enough parameters, and that the resulting zero-placement constraints, although polynomial rather than affine, can be handled by rewriting the plant as a delay-differential algebraic equation and eliminating the dependent gains. The payoff is a single-input feedback law that assigns transmission zeros at four prescribed frequencies and at the same time minimizes the spectral abscissa, so the closed loop is stable with a damping margin. A four-mass benchmark simulation shows the target mass's oscillations vanish after a short transient.

What carries the argument

The central object is the delay-differential algebraic equation (DDAE) formed by introducing slack variables for the delayed control input and delayed measured outputs, and the reinterpretation of the dynamic feedback controller as a static output feedback gain K in that augmented model. This makes the zero-location equations polynomial in K and lets the paper recycle static output feedback design machinery. The elimination step is carried by the Weinstein–Aragozajn identity: for each targeted frequency ω_k, the constraint is reduced to 1 − g^T [C_g 0]R(ω_k,K_L)^{−1}[B_11;0] = 0, assuming R is invertible; stacking the m real and imaginary parts gives the linear system P(K_L)g = Q. The dependent parameters g are then solved for as g(K_L)=P(K_L)^{−1}Q, turning the constrained problem into the unconstrained spectral-abscissa minimization, solved with a quasi-Newton BFGS method with weak Wolfe line search because the objective is nonsmooth and nonconvex.

What would settle it

Evaluate the full zero-location determinant h_k(K) at the optimizer returned for the four-mass system; if any of the four complex constraints is nonzero beyond numerical tolerance, the elimination did not preserve the original problem. The paper reports the tuned zeros are assigned correctly in the pole-zero plot, so a direct check on the unreduced determinant is the decisive test.

Watch

Extended reading notes

Core claim

The central claim is that multi-frequency non-collocated vibration suppression can be reduced to a constrained optimization problem with polynomial zero-location constraints and solved with static-output-feedback design tools, provided the feedback is enlarged with multiple fixed delays and a dynamic controller. The authors remodel the plant together with the delayed feedback and controller as a DDAE, which lets the delayed dynamic controller be reinterpreted as a static output feedback gain K. The zero-location constraints h_k(K)=0, k=1,...,m, force transmission zeros of the disturbance-to-target transfer function to lie at ±jω_k. Because these constraints are affine in a carefully chosen subset of entries of K, they can be eliminated: the dependent gains are expressed as g(K_L)=P(K_L)^{-1}Q, leaving an unconstrained nonsmooth, nonconvex optimization in the independent gains K_L, whose objective is the spectral abscissa α(K_L). In the case study, with four delays and controller orders zero through three, the closed-loop spectral abscissa reaches −0.5218 to −0.5347, and the target displacement settles to zero under a four-harmonic disturbance.

Load-bearing premise

The design assumes the matrix R(ω_k,K_L) and the elimination matrix P(K_L) remain invertible throughout the optimization; when either becomes singular, the reduced constraint set no longer represents the original zero-placement condition, so the returned controller may not put the zeros where promised.

Editorial extensions

If this is right

A single voice-coil actuator can suppress more independent harmonics than it has measured outputs, because each added delay multiplies the number of free controller parameters.
The same DDAE reformulation extends the method to multi-input settings and to additional control objectives like tracking, since the elimination leaves surplus parameters when the controller order or delay count is high.
The zero-order controller already gives α=−0.5218, and increasing controller order improves the stability margin (α=−0.5347 at order three), so dynamic feedback buys damping without needing extra delays.
Because the constraint elimination is algebraic, the design runs in existing time-delay system software and inherits its numerical treatment of infinitely many characteristic roots.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the paper leaves implicit is replacing the fixed delay set by tunable delays: treating τ_i as optimization variables could shrink the controller order needed or improve the margin, but it would break the affine structure the elimination relies on.
The robustness questions flagged in the discussion could be probed by computing the sensitivity of the assigned zeros to small parameter variations; if the zeros drift rapidly with mass or stiffness changes, the practical suppression bandwidth will be narrow even though the nominal design is exact.
The method's logic is not limited to mechanical vibration; the same zero-placement-plus-spectral-abscissa formulation applies to any system whose disturbance-to-output transfer function can be computed, such as acoustic ducts or flexible structures, as long as the delay structure is admissible.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Adrian and coauthors have extended their earlier delayed static-feedback framework to dynamic feedback controllers, using a DDAE remodeling to keep the optimization close to their previous work. The new element is real: with dynamic feedback, the zero-location constraints become polynomial rather than affine, and the paper shows how to eliminate the dependent parameters using the Weinstein–Aragozajn identity, then minimize the spectral abscissa with a nonsmooth solver. The case study, with four target frequencies, a single actuator, and only non-collocated sensors, is a good stress test, and the reported monotonic improvement in spectral abscissa as controller order increases is internally consistent.

Where the paper is soft is exactly the elimination step. Equations (13) and (16) require R(ω_k, K_L) and P(K_L) to be invertible, and the paper just states “assuming R is invertible” without analyzing when that fails. If either matrix becomes singular, the reduced constraints no longer represent the original determinant conditions h_k(K)=0. Since the controller gains are not reported, the reader cannot check whether the claimed solution actually places the four zero pairs. This is a reproducibility gap and a load-bearing one. The paper should at least add a post-hoc verification of the original constraints, and ideally discuss conditions under which the elimination is valid. The omission of controller parameters and code makes this hard to evaluate.

A second, smaller point: the benefit of dynamic feedback in the example is modest. The static controller with four delays already gives α = −0.5218, and the order-3 controller only improves this to −0.5347. That does not invalidate the method, but it means the added complexity buys little in this particular case, and the paper would be stronger with a baseline comparison or a problem where the dynamic part is essential.

On balance, this is a solid incremental contribution to the time-delay vibration-control literature. The connection to the authors’ prior work is clear and honest, and the DDAE formulation is a sensible way to handle the added complexity. It deserves a serious referee, but the revision should address the invertibility assumptions and provide enough data to reproduce the result.

Referee Report

2 major / 5 minor

Summary. The paper proposes a methodology for designing a delayed dynamic-feedback controller for multi-frequency non-collocated vibration suppression. The authors remodel the system as a delay-differential algebraic equation (DDAE) and formulate the design as a constrained optimization problem that minimizes the spectral abscissa subject to zero-location constraints, i.e., placing transmission zeros of the disturbance-to-target transfer function at prescribed imaginary frequencies. Using the Weinstein-Aragozajn identity, the constraints are eliminated by expressing 2m dependent controller gains as a function of the remaining independent parameters, yielding an unconstrained nonsmooth nonconvex optimization problem that is solved with the HANSO solver. A case study on a four-mass system with four harmonic frequencies (4, 8, 12, 16 Hz) reports negative spectral abscissa values for controller orders 0 through 3, with slightly improved values at higher orders, and a time-domain simulation shows silencing of the target-mass oscillation after the controller is activated.

Significance. The problem addressed is relevant, and the DDAE-based reformulation combined with constraint elimination is a clever extension of the authors' earlier work on delay-based controllers for single-frequency and static-feedback cases. If the zero-location constraints are indeed satisfied at the optimized solutions, the method would provide a systematic single-input design procedure for multi-frequency non-collocated vibration suppression with dynamic controllers, which is currently lacking in the literature. The paper gives a plausible derivation and reports a sensible qualitative trend of improving spectral abscissa with controller order. However, the central claim rests on a reduction step whose validity requires invertibility of the matrices R(omega_k,K_L) and P(K_L), and the paper neither analyzes when these conditions hold nor verifies numerically that the final solution satisfies the original zero-location constraints. The omission of controller parameters and the lack of a pole-zero plot also prevent independent verification of the results. These gaps are load-bearing for the stated claims, so the paper requires revision before it can be accepted.

major comments (2)

[3.4, Eqs. (13)-(16)] The equivalence between the original zero-location constraints h_k(K)=0 in Eq. (8) and the reduced equations (15)-(16) that lead to the unconstrained problem (17) depends on the invertibility of R(omega_k,K_L) and P(K_L). The text only states 'assuming R is invertible' and 'provided the matrix P(K_L) is invertible' without any analysis of when these conditions hold or any safeguard for singular cases. If R(omega_k,K_L) is singular for some K_L, the determinant h_k(K) can vanish even when the reduced scalar equation (13) is not satisfied, so the reduced feasible set is not equivalent to the original constraint set. Similarly, if P(K_L) becomes singular, the dependent gains g cannot be computed, and the unconstrained problem (17) is not defined. Since the central claim that the method assigns transmission zeros at the prescribed frequencies rests on this reduction, the paper must provide sufficient conditions for invertibility, or include a numerical mechanism to detect and avoid singular R or P during the optimization, or verify at the computed solution that the original determinants h_k(K) are zero to numerical precision and that R and P are well conditioned. Without such analysis or verification, the reported alpha values and the simulation do not establish that the designed controller actually places the transmission zeros.
[4.2, Table 2, Fig. 3] The case study does not provide the controller parameters, which are stated as omitted due to limited space, nor a pole-zero plot verifying that the closed-loop transmission zeros are located at the four prescribed frequencies. The caption of Fig. 3 claims 'the desired zeros are assigned correctly', but the figure itself is not included in the manuscript, so the reader cannot confirm that claim. Table 2 reports only the spectral abscissa values alpha, which are necessary but not sufficient: a controller that fails to place the zeros could still yield a negative alpha in the time-domain simulation because the closed-loop system might be stable even if the prescribed zeros are not exactly assigned. To support the central claim, the paper should include the explicit controller gains (at least the dependent gains g and the independent parameters K_L) or provide a supplementary file, together with a numerical check of |h_k(K)| at the optimized parameter vector, and ideally a plot of the zero locations. This would also allow the reader to assess whether the small improvements in alpha with increasing controller order (from -0.5218 to -0.5347) are meaningful and not merely numerical noise.

minor comments (5)

[3.4, Eq. (13)] Please provide a reference for the Weinstein-Aragozajn identity, as it may not be familiar to all readers of the journal.
[3.4, Eqs. (15)-(16)] The construction of the matrix P(K_L) and the vector Q in Eq. (16) is not explained in detail; please clarify the dimensions and how P is assembled from the vector z(omega_k,K_L) defined just below Eq. (15).
[4.2, after Table 2] The sentence 'the resulting optimum solution achieved with the higher-order controller can be no lower than the optimum computed for the previous lower-order controller' is logically reversed; since alpha is minimized, a higher-order controller should achieve an optimum that is no larger (i.e., no worse) than the lower-order optimum.
[3.2, Eq. (7)] Equation (7) contains a typographical error: the determinant expression is not properly typeset, with '−~B1K~C1 − ~B' followed by '~C 0' in a way that obscures the Rosenbrock matrix structure; please rewrite the matrix explicitly with brackets and define all symbols.
[4.1] The statement 'we would require at least eight free controller parameters, one for each zero' is imprecise: each complex conjugate zero pair requires two real equations, so eight free parameters for four pairs is the correct count, but the phrasing should be clarified.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the zero-location constraints are imposed design goals, and the cited elimination is an algebraic reformulation, not a load-bearing self-citation.

full rationale

Walked the derivation chain: the paper's contribution is a design method, not an empirical prediction. The zero-location constraints (8) are imposed design goals, and the elimination (13)-(16) is an algebraic reformulation (Weinstein-Aragozajn) of those constraints under stated invertibility assumptions; it does not reduce to the result by construction. The spectral abscissa minimization (17) is an optimization objective, and Table 2 reports the optimized objective, not a fitted prediction. The simulation in Section 4 confirms that the designed controller places the zeros, which is a consistency check; the paper explicitly defers experimental validation to future work. The elimination procedure is cited from the authors' prior work (Saldanha et al. 2022, 2024), but the necessary equations are given in the text and the cited procedure is a parameter-free algebraic method, so the self-citation is not load-bearing in a circular sense. The only caveats are the invertibility assumptions on R(omega_k, K_L) and P(K_L), which are correctness/robustness limitations rather than circularity. No step equates a fitted input with a prediction or imports a uniqueness theorem.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The method relies on standard control-theory assumptions (zero assignment for harmonic suppression), on the DDAE remodeling borrowed from prior work, and on the affinity/invertibility conditions that make constraint elimination valid. The only hand-chosen numerical inputs are the four fixed delays and the controller order; no new physical entities are introduced.

free parameters (2)

Fixed output delays τ_i = [0.05, 0.1, 0.15, 0.2] s
Chosen by hand as four equi-spaced delays; the paper says the interval is selected based on the signal frequencies, so the reported results may depend on this unoptimized choice.
Controller order nc = 0,1,2,3
The dynamic controller order is increased manually in successive optimization runs; the improvement from nc=1 to nc=0 is zero, suggesting a heuristic rather than systematic selection.

assumptions (4)

domain assumption Suppressing a harmonic disturbance at frequency ωk is equivalent to placing a transmission zero pair of the transfer function from fd to z at ±jωk.
Used in Section 3.2 to define the constraints h_k(K)=0; this is a standard control-theory assumption.
domain assumption Introducing slack variables and rewriting the system as (5) with u=K y preserves the relevant transfer function and spectrum.
Invoked in Section 3.1 by reference to Gumussoy and Michiels (2011); not proved in this paper.
domain assumption There exists a selection of 2m dependent parameters from a single row or column of K such that the zero-location constraints are affine in those parameters, enabling elimination (16).
Assumed in Section 3.4 by reference to Saldanha et al. (2022); if false or if P(K_L) is singular, the unconstrained reformulation is invalid.
domain assumption The matrix R(ω_k, K_L) in Eq. (13) is invertible for each targeted frequency.
Required for Eq. (13); the paper explicitly assumes invertibility without a condition or remedy.

invented entities (1)

Slack variables ζ_u and ζ_y
purpose: Augment the state to represent delayed input and output signals as state delays in a DDAE, enabling a static-like output feedback formulation.
Mathematical auxiliary variables introduced in Section 3.1; they have no physical counterpart and are not independently observable.

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Cite this review

Pith. "Pith review of Delayed dynamic-feedback controller design for multi-frequency vibration suppression." pith.science (2026). https://pith.science/paper/WOMMB3SB

@misc{pith2026250516939,
  author       = {Pith},
  title        = {Pith review of: Delayed dynamic-feedback controller design for multi-frequency vibration suppression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WOMMB3SB}},
  note         = {Machine review of arXiv:2505.16939}
}

read the original abstract

We present a methodology for designing a dynamic controller with delayed output feedback for achieving non-collocated vibration suppression with a focus on the multi-frequency case. To synthesize the delay-based controller, we first remodel the system of equations as a delay-differential algebraic equation (DDAE) in such a way that existing tools for design of a static output feedback controller can be easily adapted. The problem of achieving non-collocated vibration suppression with sufficient damping is formulated as a constrained optimization problem of minimizing the spectral abscissa in the presence of zero-location constraints, with the constraints exhibiting polynomial dependence on its parameters. We transform the problem into an unconstrained one using elimination, following which we solve the resulting non-convex, non-smooth optimization problem.

Figures

Figures reproduced from arXiv: 2505.16939 by the authors.

**Figure 2.** Block diagram of the dynamic controller with [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Pole-zero plot showing the rightmost eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Multi-harmonic disturbance force fd(t) (top) acting on mass m2, controller output u(t) (middle) and displacement x1(t) of target mass m1 (bottom). Controller is turned on at time t = 5 s, following which, after transient period, the oscillation of target mass is silenc…

Discussion (0). Continue with ORCID to comment.

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4 extracted references

[1]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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